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This is the first time I ask a question on Mathoverflow, so I apologize in advance if it is not suitable/a duplicate/otherwise inappropriated.

I am thinking about Voevodsky's category of motives and I realized that in his presheaves with transfers formalism pullbacks for Chow groups are defined for arbitrary maps of smooth schemes. Precisely if $f:X\to Y$ is a map of smooth schemes and $\alpha\in CH_*(Y)$ he defines

$$f^*\alpha = (pr_1)_*(\Gamma_f \cdot (pr_2)^*\alpha)$$

where $pr_1,pr_2$ are the projection maps from $X\times Y$ and $\Gamma_f$ is the closed subscheme of $X\times Y$ determined by the graph of $f$ (note that $(pr_1)_*$ is well defined more or less because the restriction of $pr_1$ to $\Gamma_f$ is an isomorphism).

After researching a bit I found a paper by Bloch ("Algebraic cycles and Higher K-theory") he seems to define the pullback for all maps with smooth target. However in classical treatments of intersection theory I've only seen $f^*$ defined for flat maps.

Q: In what generality is the pullback of cycle classes defined?

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  • $\begingroup$ just a quick comment: when you try and define pullback for non flat maps some smoothness/regularity comes in as you need (as you wrote) $\Gamma \cdot - $, ie an intersection product. $\endgroup$ Mar 17, 2015 at 20:44
  • $\begingroup$ Yes, if we want to use that definition. I'm interested to know in which generality we can define a reasonably well-behaved ``pullback'', not in which generality Voevodsky's definition works. $\endgroup$ Mar 17, 2015 at 20:49
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    $\begingroup$ well, if instead of working with Chow groups you work with $K_0(Coh))$ (the Grothendieck group, together with the dimension/$\gamma$/coniveau/whateverscalled filtration) one can define pullback of "cycles" as long as the map $f\colon X \to Y$ is $Tor$-finite or, in other words, the left derived functor $Lf^*$ is bounded. Surely, smoothness of the target ensures it for any $f$ (as any coherent sheaf is quasi-isomorphic to a bounded complex of vector bundles). $\endgroup$ Mar 17, 2015 at 21:08
  • $\begingroup$ That is very interesting: I should have thought immediately of the K-theoretic analogue... It would be very interesting if Tor-finiteness were in fact what was needed for defining a pullback on Chow groups too $\endgroup$ Mar 17, 2015 at 21:12
  • $\begingroup$ there is a comparison map between the associated graded of $K_0$ and Chow, which is an isomorphism up to torsion. $\endgroup$ Mar 17, 2015 at 21:15

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You have to distinguish between pullbacks of cycles, pullbacks on Chow groups, and pullbacks of relative cycles.

  • You cannot always pull back cycles. If $f: Y\to X$ is a morphism of (arbitrary) schemes and $Z\subset X$ is an elementary cycle, $f^*(Z)$ is defined provided that $Z$ is ``in good position with respect to $f$'', which simply means that $f^{-1}(Z)$ has the same codimension in $Y$ as $Z$ had in $X$; this is always true if $f$ is flat. The formula for $f^*(Z)$ is the one you wrote, where the intersection uses Serre's multiplicities.

  • Pullbacks on Chow groups are defined at least for arbitrary maps between smooth schemes over fields (see below for various generalizations). This is because of Chow's moving lemma, which says that you can always replace a cycle by a rationally equivalent one which is in good position with respect to a given map.

  • If $X$ is of finite type over $S$ regular, an equidimensional relative cycle on $X/S$ is a cycle on $X$ which is equidimensional and dominant over (a connected component of) $S$. For instance a finite correspondence from $X$ to $Y$ over $S$ is such a cycle on $X\times_SY/X$, which is moreover finite over $X$. Equidimensional relative cycles can be pulled back along any map $T\to S$. There is a notion of relative cycle which need not be equidimensional and these notions can be extended to singular schemes, but it gets technical, see Suslin-Voevodsky http://www.math.uiuc.edu/K-theory/0035/susvoe2.pdf or Cisinski-Déglise http://arxiv.org/pdf/0912.2110v3.

With hard work, the functoriality on Chow groups can be extended to Bloch's cycle complexes $z^r(X,*)$, and hence to higher Chow groups. This was done by Bloch for smooth schemes over a field, by Levine for smooth schemes over a Dedekind domain (https://www.uni-due.de/~bm0032/publ/ChowMovLemFinal.pdf), and recently by Spitzweck for smooth schemes over variable Dedekind domains (http://arxiv.org/pdf/1207.4078.pdf). A huge advantage of Voevodsky's definition of the motivic complexes is that the functoriality is apparent, since these complexes are defined in terms of equidimensional cycles. At the end of the day Voevodsky's complex is equivalent to Bloch's, but that requires a moving lemma (which is known for fields but not for more general Dedekind domains). The basic idea for the comparison is that, up to controlled rational equivalence, a codimension $r$ cycle on $X$ is as good as a codimension $r$ cycles on $X\times\mathbb{A}^r$ (by homotopy invariance), and any such can be moved to be of relative dimension zero over $X$.

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  • $\begingroup$ Do you have a reference for the statement in the first bullet? That would be a good answer to my question: math.stackexchange.com/questions/1896395/… $\endgroup$ Aug 21, 2016 at 18:26
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    $\begingroup$ @OliverE.Anderson I was very sloppy in that first bullet point, sorry: $X$ and $Y$ need to be over a base $S$, with $X/S$ flat, for the formula $\Gamma_f\cdot \operatorname{pr}_2^*(Z)$ to make any sense; and then one needs conditions for Serre's multiplicities to be defined. Probably even stronger conditions are needed for this to be a well-behaved operation. According to Appendix B in Spitzweck's paper, everything seems fine when $S$ regular and $X$ and $Y$ smooth over $S$. $\endgroup$ Aug 28, 2016 at 17:03

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